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Chapter 5

# Chapter 5 - Chapter5: (5.1 Definition:. y = b x b > 0 b 1...

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Chapter 5: Exponential and Logarithmic Functions Exponential Functions (5.1) Definition: Let b denote a positive constant other than 1. The exponential function with base b is defined by the equation y = b x , b > 0, b 1 Ex: Note: y = x 2 , y = x are not examples of exponential functions. The base has to be a constant in order to qualify as one. Sketch: y = 2 x Sketch: y = ( 1 2 ) x Domain: Domain: Range: Range: Difference between the two graphs:

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Now on I will not plot points to refer to a standard exponential function graph. Instead the shape as above will be drawn for y = b x , b > 0, b 1 Now Sketch : y = 2 x and compare it to the graph of y = 2 x Domain: Range: Difference between the two graphs: Ex: Sketch y = 2 x 3 without plotting points by translating and reflecting appropriately the basic graph.
Ex: Sketch y = 2 x , y = 3 x , y = ( 3 2 ) x , y = 4 x on the same set of axes. How does the change in the base value affect the shape of the general look of an exponential graph? Properties of Graphs of Exponential Functions: Let f ( x ) = b x , b > 0, b 1. Then the graph of f(x): 1) Is continuous for all real numbers 2) Has no sharp corners 3) Passes through the point (1, 0) 4) Lies above the x‐axis which is a horizontal asymptote either as x → ∞ or x - but not both 5) Increase as x increases if b>1, decreases as x increases if 0<b<1 6) f(x) is one to one, that is intersects any horizontal line at most once.

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Solve the following exponential equations: Ex: Solve for t: ( 1 3 ) 2 t = 9 t 6 Ex: Solve for x: 27 2 x + 4 = 9 4 x Properties of real number exponents for b x , b > 0, b 1 1) If x is real, b x is a unique real number example: 3 1.2 2) If x is an integer, b x is known example: 3 2 3) If x is rational, b x can be expressed as a radical example: 3 2 3 = 3 2 3 4) If x is irrational, approximate b x with b r where r x example 3 2 3 1.2 Remember: All the properties related to exponents will hold true for 1‐4
The Exponential Function y = e x (5.2) Note: The base e is called the natural base and for our purpose e 2.71 e is an irrational number which is approximately equal to 2.7182818284…….. The base e is sometimes easier to use than the other bases. Graph y = b x , b > 0 by remembering the general look of an exponential equation Domain: Range: y‐intercept: x‐intercept: Asymptote: Think of the base b as the constant e. Ex: Graph y = e x + 1 by specifying the domain, range, intercepts and asymptotes

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How do we define e to be 2.71828……. Let’s investigate the values of function f ( x ) = 1 + ( 1 x ) x as x gets larger and larger x f ( x ) = 1 + ( 1 x ) x 10 10 2 10 3 10 4 .
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